Enrique Alonso

Completeness: from Gödel to Henkin

This paper focuses on the evolution of the notion of completeness in contemporary logic. We discuss the differences between the notions of completeness of a theory, the completeness of a calculus, and the completeness of a logic in the light of Gödels and Tarskis crucial contributions.We place special emphasis on understanding the differences in how these concepts were used then and now, as well as on the role they play in logic. Nevertheless, we can still observe a certain ambiguity in the use of the close notions of completeness of a calculus and completeness of a logic. We analyze the state of the art under which Gödels proof of completeness was developed, particularly when dealing with the decision problem for first-order logic. We believe that Gödel had to face the following dilemma: either semantics is decidable, in which case the completeness of the logic is trivial or, completeness is a critical property but in this case it cannot be obtained as a corollary of a previous decidability result. As far as first-order logic is concerned, our thesis is that the contemporary understanding of completeness of a calculus was born as a generalization of the concept of completeness of a theory. The last part of this study is devoted to Henkins work concerning the generalization of his completeness proof to any logic from his initial work in type theory.

Diagonalisation and Church's Thesis: Kleene's Homework

In this paper we will discuss the active part played by certain diagonal arguments in the genesis of computability theory. 1 In some cases it is enough to assume the enumerability of Y while in others the effective enumerability is a substantial demand. These enigmatical words by Kleene were our point of departure: When Church proposed this thesis, I sat down to disprove it by diagonalizing out of the class of the λ–definable functions. But, quickly realizing that the diagonalization cannot be done effectively, I became overnight a supporter of the thesis. (1981, p. 59) The title of our paper alludes to this very work, a task on which Kleene claims to have set out after hearing such a remarkable statement from Church, who was his teacher at the time. There are quite a few points made in this extract that may be surprising. First, it talks about a proof by diagonalization in order to test—in fact to try to falsify—a hypothesis that is not strictly formal. Second, it states that such a proof or diagonal construction fails. Third, it seems to use the failure as a support for the thesis. Finally, the episode we have just described took place at a time, autumn 1933, in which many of the results that characterize Computability Theory had not yet materialized. The aim of this paper is to show that Church and Kleene discovered a way to block a very particular instance of a diagonal construction: one that is closely related to the content of Churchs thesis. We will start by analysing the logical structure of a diagonal construction. Then we will introduce the historical context in order to analyse the reasons that might have led Kleene to think that the failure of this very specific diagonal proof could support the thesis. This is a joint paper. We have both attempted to add a small piece to an amazing historical jigsaw puzzle at a juncture we feel to be appropiate. In the paper by Manzano 1997 the aforementioned words by Kleene were quoted, and since then several logicians, Enrique Alonso first and foremost, have questioned her on this issue. Here we both submit our reply. (1999, pp. 249--273)

A note on Visions of Henkin

In 2007, just a few months after Leon Henkin passed away, we presented a paper at a small Symposium on Lógica, Filosofía del Lenguaje y de la Lógica organized at Seville University. It was published in the proceedings of the event as “Leon Henkin” in A. Nepomuceno et al. (eds.) (2007), Lógica, Filosofía del Lenguaje y de la Lógica. Sevilla: Mergablum. In this paper we offer a very preliminary stage of our research on Henkin’s life and work. In 2012, on the occasion of István Nemeti’s 70th birthday, we participated in the the First International Conference on Logic and Relativity with “Our Beloved Leon Henkin”. The present Synthese paper, “Visions of Henkin”, is related to that contribution. While preserving the basic biographical items of the preceding Spanish paper, “Visions of Henkin” offers more detailed information with respect to Henkin’s academic positions. It also goes deeper into Henkin’s influences during his formation at Columbia, specially in those aspects having to do with the reading of some Quine papers; in particular, the paper where Quine proves completeness for propositional logic. We also mention the role played by Frege’s readings during this same period. The subsection devoted to the relation of Leon Henkin with Spain has been left out. With respect toHenkin’s specific contributions,wehave addedmore details concerning the role played by some concepts introduced for the theory of types in his proof of the first-order completeness theorem. We also added the relation the completeness proof

Visions of Henkin

Leon Henkin (1921–2006) was not only an extraordinary logician, but also an excellent teacher, a dedicated professor and an exceptional person. The first two sections of this paper are biographical, discussing both his personal and academic life. In the last section we present three aspects of Henkin’s work. First we comment part of his work fruit of his emphasis on teaching. In a personal communication he affirms that On mathematical induction, published in 1969, was the favourite among his articles with a somewhat panoramic nature and not meant exclusively to specialists. This subject is covered in the first subsection. Needless to say that we also analyse Henkin’s better known contribution: his completeness method. His renowned results on completeness for both type theory and first order logic were part of his thesis, The Completeness of Formal Systems, presented at Princeton in 1947 under the advise of Alonzo Church. It is interesting to note that he obtained the proof of completeness for first order logic readapting the argument for the theory of types. The last subsection is devoted to philosophy. The work most directly related to philosophy is an article entitled: Some Notes on Nominalism which appeared in the Journal of Symbolic Logic in 1953. Unfortunately, we are not covering his contribution to the field of cylindric algebras. As a matter of fact, Henkin spent many years investigating algebraic structures with Alfred Tarski and Donald Monk, among others.

Henkin’s Theorem in Textbooks

Our aim in this paper is to examine the incorporation and acceptance of Henkin’s completeness proof in some textbooks on classical logic. The first conclusion of this paper is that the inclusion of Henkin’s completeness proof into the standards of Logic was neither quick nor easy. Surprising as it may seem today, most of the textbooks published in the 1950s did not include a section for this proof, nor presented it in any way. A question we should try to answer is at what moment does Henkin’s proof of completeness for first order logic begin to be considered as a part of the standards of elementary logic. This point brings us to a discussion on the way in which the specific gains of Henkin’s proof have been assessed in literature. The possibility of using Henkin’s methods in a wide variety of formal systems made completeness a general property belonging to foundations of logic, leaving the realm of model theory for quantification languages where it was previously located.